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THE EXISTENCE OF HYPERSURFACES OF CONSTANT GAUSSCURVATURE WITH PRESCRIBED BOUNDARY

BO GUAN AND JOEL SPRUCK

1. Introduction

In this paper we are concerned with the problem of finding hypersurfaces of con-stant Gauss-Kronecker curvature (K-hypersurfaces) in R

n+1 (n ≥ 2) with prescribedboundary: given a disjoint collection Γ = Γ1, . . . ,Γm of closed smooth embedded(n − 1) dimensional submanifolds of R

n+1, decide whether there exist (immersed)K-hypersurfaces M in R

n+1 with ∂M = Γ. Locally this problem reduces to ques-tions concerning Monge-Ampere type equations and we seek solutions for which theresulting equation is elliptic. This means that we must confine ourselves to the classof locally strictly convex hypersurfaces, i.e. those whose principal curvatures are allpositive. Such hypersurfaces locally lie on one side of their tangent planes at anypoint but need not do so globally as they have non-empty boundary.

Finding hypersurfaces with prescribed curvature and boundary has been a majorchallenge in geometric analysis because of the highly nonlinear nature of the prob-lem and the lack of variational methods. Beginning around 1980, some success wasachieved due to breakthroughs in the theory of Monge-Ampere equations and generalfully nonlinear equations, but only for hypersurfaces which are globally graphs of func-tions over domains with geometric restrictions (e.g., strictly convex domains). Thiswas the case even for K-hypersurfaces where the only general existence results wereconsequences of the existence theory for Monge-Ampere equations (see [6], [20], [23]),and was restricted to strictly convex domains. This means that the resulting surfacesmust be simply connected graphs, a very strong restriction geometrically.

The first idea that a more general result was possible came in the paper of Hoffman-Rosenberg-Spruck [18] and subsequently such a general result was developed in [12]and [11]. In these papers, the authors proved an essentially optimal existence theoremfor Monge-Ampere equations in domains of arbitrary geometry and thus the limit of

Research of the first author was partially supported by NSF Grant DMS-9626722. Research ofthe second author was partially supported by NSF Grant DMS-9703154.

1

2 BO GUAN AND JOEL SPRUCK

our understanding of K-hypersurfaces with boundary was reached, as far as globalgraphs (including multi-sheeted radial graphs) are concerned. This theory alreadyled to striking geometric applications [12], [28], [8].

To solve the problem in its full parametric generality seemed to require substan-tial new techniques. A necessary condition for Γ to bound a locally strictly convexhypersurface is that its second fundamental form (as a submanifold of R

n+1) is non-degenerate everywhere. This however, is not a sufficient condition; Rosenberg [27](seealso [10]) shows there are topological obstructions. It is natural to seek geometric con-ditions that guarantee the existence of locally strictly convex K-hypersurfaces span-ning a given Γ. Based on the results in [12], the second author [29] made the followingconjecture: Γ must bound an immersed K-hypersurface if it bounds a locally strictlyconvex immersed hypersurface. The first main result of the present paper settles thisconjecture affirmatively. More precisely, we will prove

Theorem 1.1. Assume that there exists a locally convex immersed hypersurface Σ inRn+1 with ∂Σ = Γ and KΣ ≥ K everywhere, where K is a positive constant. Suppose,

in addition, that Σ is C2 and locally strictly convex along its boundary. Then thereexists a smooth (up to the boundary) locally strictly convex immersed hypersurface Mwith ∂M = Γ such that KM ≡ K. Moreover, M is homeomorphic to Σ.

We note that this is a huge jump in generality from our previous results in [12] as itdeals with general immersed K-hypersurfaces and not just graphs (or radial graphs).Because of the presence of boundary, locally convex surfaces can be very complicated.In particular, in Theorem 1.1 M need not be embedded even if Σ is embedded.

It is also important to understand hypersurfaces of vanishing Gauss curvature.These hypersurfaces are clearly related to convex hulls of codimension 2 submanifoldsin space. Our second main result in this article is the following

Theorem 1.2. Suppose Γ bounds a locally convex hypersurface which is C2 and lo-cally strictly convex along its boundary. Then there exists a locally convex hypersur-face M of Gauss curvature KM ≡ 0 with ∂M = Γ, and M is of class C1,1 up tothe boundary. Moreover, for any interior point p ∈ M , all the extreme points of the(intrinsic) component of M ∩ TpM containing p lie on ∂M , where TpM denotes thetangent plane of M at p. In particular, if Γ is extreme, i.e., Γ lies on the boundaryof its convex hull, then M coincides with part of the boundary of the convex hull of Γand, therefore, is globally convex.

HYPERSURFACES OF CONSTANT GAUSS CURVATURE 3

The C1,1 regularity in Theorem 1.2 is optimal for hypersurfaces of vanishing Gausscurvature, as shown by counterexamples (see [7]). We also remark that Theorem 1.2does not hold without the assumption that Γ bounds a locally convex hypersurfacewhich is locally strictly convex near its boundary. Ghomi [8] has constructed a smoothextreme Jordan curve γ in R

3 with the properties that (a) γ bounds a convex surfaceof vanishing Gauss curvature which is not C1,1, (b) γ does not bound any locallystrictly convex surface, and (c) γ does not bound any locally convex surface of classC1,1 with vanishing Gauss curvature.

As a consequence of Theorem 1.2 we have

Corollary 1.3. Suppose Γ is extreme and let Σ be a locally convex hypersurface with∂Σ = Γ. If Σ is C2 up to the boundary and locally strictly convex in a neighborhoodof its boundary, then the interior of Σ lies strictly outside the convex hull of Γ.

We remark that such a hypersurface need not be globally convex, nor embedded. Asomewhat stronger version of Corollary 1.3 has been proven by Alexander-Ghomi [1].In [8], Ghomi made the following conjecture: every compact connected hypersurface ofpositive curvature with connected extreme boundary is embedded and its interior liesoutside the convex hull of its boundary. We see that Corollary 1.3 settles affirmativelypart of this conjecture. On the other hand, we will construct an example which showssuch a hypersurface may fail to be embedded. Furthermore, using the bridge principleof Hauswirth [16], we will show there exist smooth K-surfaces in R

3, with connectedextreme boundary, which are not embedded.

Suppose Γ is extreme and let HΓ be the boundary of its convex hull. Theorem 1.2indicates that if Γ bounds a locally convex hypersurface which is C2 (up to theboundary) and locally strictly convex in a neighborhood of its boundary, then oneof the components of HΓ \ Γ must be C1,1 up to the boundary. However, as we willshow by an example, the other components may have interior singularities. A resultof Ghomi [8] states that every component of HΓ \ Γ is C1,1 up to the boundary if Γis strictly convex, i.e., through every point of Γ there passes a (global) supportinghyperplane with first order contact.

Hypersurfaces of vanishing Gauss curvature are closely related to the homogeneousdegenerate Monge-Ampere equation

detD2u = 0.(1.1)


In general, the Dirichlet problem for (1.1), even with smooth boundary data, does nothave C2 solutions, as shown by an example of Urbas (see [7]). Under suitable regu-larity assumptions on the boundary data, the interior and global C1,1 regularity wasestablished by Trudinger-Urbas [31] and Caffarelli-Nirenberg-Spruck [7], respectively,for strictly convex domains. Later the first author [11] extended the global regularityresult of [7] to non-convex domains. These regularity results will play important rolein our proof of Theorem 1.2. For more general (non-homogeneous) degenerate Monge-Ampere equations, the C1,1 regularity has been studied by Caffarelli-Kohn-Nirenberg-Spruck [5], Hong [19], Krylov [24], P.-F. Guan [13] and Guan-Trudinger-Wang [14],etc.

A major difficulty in proving Theorems 1.1 and 1.2 lies in the lack of global co-ordinate systems to reduce the problem to solving certain boundary value problemfor Monge-Ampere type equations. To overcome this difficulty, we adopt a Perronmethod to deform (lift) Σ into a K-hypersurface by solving the Dirichlet problem forthe Gauss curvature equation (2.1) locally. This approach, while classical for PDE’s,requires substantial technical work as we are dealing with general locally convex hy-persurfaces in space. A key ingredient, among others, is an a priori estimate for thelocal Lipschitz constants (C0,1 norms) of locally convex hypersurfaces spanning Γ.This is established in section 3 where we also derive a priori estimates for the lowerand upper bounds of principal curvatures of locally strictly convex K-hypersurfacesspanning Γ. The Perron method is carried out in section 4 where we define the defor-mation space L of liftings of Σ and construct M as the limit of a suitable sequence ofhypersurfaces in L. In section 5, we study the regularity of the resulting hypersurfaceconstructed in section 4, to complete the proofs of Theorems 1.1 and 1.2. Finally, insection 6 we prove Corollary 1.3 and construct an extreme curve in R

3 which boundsa locally strictly convex K-surface with self-intersection and for which the boundaryof its convex hull has interior singularities.

For general Monge-Ampere equations, there is a vast literature, with fundamen-tal work being done by Pogorelov, Cheng-Yau, Lions, Ivochkina, Krylov, Caffarelli-Nirenberg-Spruck, Trudinger, Urbas and others in the 1970-1980’s and more recentregularity results by Caffarelli. For further references the reader is referred to [9], [15]and the expository article [25].

After this article was completed, we learned that Trudinger-Wang [32] also provedTheorem 1.1 at about the same time.


2. Notation and Preliminaries

Let Φ : Σn0 → R

n+1 be an immersion where Σ0 is a manifold of dimension n ≥ 2with boundary ∂Σ0 which may be empty. We will often identify Φ with its imageM := Φ(Σ0) and call M a hypersurface of R

n+1. Similarly, the boundary of M , ∂Σ,means the immersion Φ : ∂Σ0 → R

n+1. When we consider a point p ∈ M , it shouldbe understood as one of its preimages in M0. For a subset U of R

n+1, Σ ∩p U willdenote the component of Σ ∩ U that contains p, that is, Σ ∩p U = Φ(U0) where U0 isthe component of Φ−1(Σ∩U) that contains the point identified to p in Φ−1(p) ⊂M0.In this paper, all hypersurfaces in R

n+1 we consider are assumed to be connected,orientable and compact with or without boundary. Unless otherwise indicated, if twohypersurfaces have the same boundary, they are assumed to be oriented in such a waythat they induce the same orientation on the boundary.

Let Σ be a C2 hypersurface in Rn+1. We will useKΣ, νΣ and dΣ to denote the Gauss

curvature, the unit normal vector field, and the extrinsic diameter of Σ, respectively.The orientation of Σ is assumed to be consistent with νΣ which is continuously definedon entire Σ. At a point on Σ the Gauss curvature KΣ is the product of the principalcurvatures which are the eigenvalues of the second fundamental form of Σ computedwith respect to νΣ. We denote by κmin[Σ] and κmax[Σ] the minimum and maximum,respectively, of all principal curvatures of Σ. We say Σ is locally convex (locally strictlyconvex) if κmin[Σ] ≥ 0 (κmin[Σ] > 0, respectively).

We will also need to consider hypersurfaces with less regularity. In general, ahypersurface Σ in R

n+1 is said to be locally convex if at every point p ∈ Σ thereexists a neighborhood which is the graph of a convex function xn+1 = u(x), x ∈ R

n,with s suitable coordinate system in R

n+1, such that locally the region xn+1 ≥ u(x)always lies on a fixed side of Σ. (Note that Σ is assumed to be orientable so it has twosides; for convenience we will refer to the inner side as the one facing xn+1 ≥ u(x).)The latter requirement that the region xn+1 ≥ u(x) lie on one fixed side of Σ is toensure that the local convexity at each point is consistent with a fixed orientation;see [1] for a detailed discussion. Note that a locally convex hypersurface is necessarilyof class C0,1 in the interior.

For a locally convex hypersurface Σ which is not necessarily C1, νΣ is understoodas the Gauss map from Σ to the subsets of S

n: for a point p ∈ Σ, νΣ(p) is the set ofall unit normal vectors of local supporting hyperplanes of Σ at p. For convenience, wewill say νΣ has a certain property of a vector if every element of νΣ has that property.For the definition in weak sense of Gauss curvature we refer to [26]. According to


Caffarelli [2], if Σ is the graph of a locally convex function xn+1 = u(x) over a domainΩ in R

n then KΣ = K if and only if u is a viscosity solution of the Gauss curvatureequation

det(uij) = K(1 + |∇u|2)n+2

2 in Ω.(2.1)

One can similarly interpret the meanings of KΣ ≤ K and KΣ ≥ K. We will needthe following existence result which follows from, for example, Theorem 1.1 of [11] byapproximation.

Theorem 2.1. Let Ω be a bounded domain in Rn with ∂Ω ∈ C0,1. Suppose there

exists a locally convex viscosity subsolution u ∈ C0,1(Ω) of (2.1), i.e.,

det(uij) ≥ K(1 + |∇u|2)n+2

2 in Ω,

where K ≥ 0 is a constant. Then there exists a unique locally convex viscosity solutionu ∈ C0,1(Ω) of (2.1) satisfying u = u on ∂Ω.

3. A priori estimates and compactness

In this section we prove some important local properties of locally convex hyper-surfaces with boundary. Throughout the section, let Σ and M be locally convexhypersurfaces in R

n+1 with ∂Σ = ∂M and assume that there exists a fixed constantδ > 0 such that the hypersurface

Σδ := x ∈ Σ : distΣ(x, ∂Σ) < δ

is C2 up to the boundary and locally strictly convex, where distΣ denotes the intrinsicdistance on Σ. We furthermore assume that M locally lies on the inner side of Σalong the boundary and any neighborhood of ∂M in M does not intersect Σδ in theinterior. By this we mean that νΣ(p) · (q− p) > 0 for all p ∈ Σδ and q ∈M near ∂M .In particular, both Σ and M locally lie on the same side of the tangent plane to Σ atany point of ∂Σ. Let Π denote the second fundamental form of ∂Σ (as a submanifoldof R

n+1). The main result, which plays a key role in our proof of Theorems 1.1 and1.2, may be stated as follows.

Theorem 3.1. At every point on M , locally M can be represented as the graph ofa convex function u defined in a domain Ω ⊂ R

n of a fixed lower bound in size(depending only on δ, κmin[Σδ], κmax[Σδ], max∂Σ |Π| and dM ) such that

‖u‖C0,1(Ω) ≤ C1(3.1)


where C1 depends on δ, κmin[Σδ], κmax[Σδ], max∂Σ |Π| and dM .

Proof. Step 1. We first note the following simple fact. Let p be an arbitrary pointon ∂Σ and X a unit tangent vector to ∂Σ at p . Since Σ is strictly locally convexnear ∂Σ, we have

νΣ(p) ·Π(X,X) ≥ κmin[Σδ] > 0.(3.2)

Throughout this proof let β = 12 sin−1(κmin[Σδ]/max∂Σ |Π|). Then

νΣ(p) ·Π(X,X)|Π(X,X)|

≥ κmin[Σδ]max∂Σ |Π|

= sin 2β > 0.(3.3)

Thus the angle between νΣ(p) and Π(X,X) does not exceed π2 − 2β.

Now, for a fixed point p ∈ ∂Σ, we take p to be the origin and choose a coordinatesystem of R

n+1 such that en and en+1 are normal to ∂Σ at p and

νΣ(p) = en cosβ + en+1 sinβ.(3.4)

Here ek is the unit vector in the positive xk-axis direction (1 ≤ k ≤ n+ 1). For laterreference we will call this the special coordinate system at p. It follows that Σ (locallyat p) can be represented as the graph of a strictly convex function xn+1 = u(x) overa domain Ω′ with a lower bound in size which depends on δ, β and κmax[Σδ]. Inparticular, ∂Σ is locally a graph over a portion, which we denote as Γ′, of ∂Ω′. By(3.3) and (3.4), the angle between en and Π(X,X) does not excess π

2 − β, that is

en ·Π(X,X) ≥ |Π(X,X)| sinβ ≥ κmin[Σδ] sinβ(3.5)

for any unit tangent vector X to ∂Σ at p. Consequently, (possibly after a rotation ofthe (x1, . . . , xn−1) coordinates) we may represent Γ′ as a graph

xn = ϕ(x′) ≡n−1∑i=1

aix2i + o(|x′|2), x′ = (x1, . . . , xn−1) ∈ R

n−1(3.6)

for some constants ai, 1 ≤ i ≤ n− 1, satisfying

0 < κmin[Σδ] sinβ ≤ ai ≤ max∂Σ|Π|, 1 ≤ i ≤ n− 1.(3.7)

By shrinking the size of Ω′ as necessary, we may assume Ω′ = ϕ < xn < 2r forsome uniform constant r > 0.

Let v be the convex function defined on Ω′ by

v(x) = supL(x) : L is an affine function, L ≤ u on Γ′.(3.8)


We have

u ≤ v ≤ maxΓ′

u in Ω′, v = u on Γ′

and

LipΩ′(v) ≤ maxΓ′|∇u|+ C

where LipΩ′(v) denotes the Lipschitz coefficient of v on Ω′.By the local convexity of M we have νM ·Π(X,X) ≥ 0 for any tangent vector X

to ∂Σ. From (3.3) we see that the angle between νM and νΣ at any point on ∂Σ doesnot exceed π − 2β. Therefore,

νM (p) · en+1 ≥ sinβ.

That is, the angle between νM (p) and en+1 does not exceed π2 − β. Consequently, M

locally (near p) can be represented as the graph of a convex function xn+1 = u(x).Since M is locally convex, we see that u is defined on a smooth strictly convex domainΩp satisfying

ϕ < xn < r ⊂ Ωp ⊂ ϕ < xn < 2r(3.9)

with

u ≤ u ≤ v in Ωp and LipΩp(u) ≤ C(3.10)

where C depends on r and ‖u‖C1(Ω′). For later reference we set

∂′Ωp := xn = ϕ|0 ≤ xn ≤ r ⊂ ∂Ωp

and by Γ(p) the graph of u over ∂′Ωp. Note that Γ(p) ⊂ ∂M .

Step 2. Next, let q be an interior point of M . We will consider two different cases.We first assume that there exists a hyperplane P through q, which either is a localsupporting hyperplane or is transversal to M at q, such that

Ut ∩ ∂M = ∅ for all t > 0 sufficiently small(3.11)

where Ut = M ∩q z ∈ Rn+1 : (z− q) · νP ≤ t. We first note the following fact which

will often be used in the sequel without being explicitly referred to.

Lemma 3.2. Suppose s > 0 such that (3.11) holds for all nonnegative t ≤ s. ThenUs is transversal to Ps := z ∈ R

n : (z − q) · νP = s.


Proof. We note that Ut is transversal to Pt for all t > 0 sufficiently small. Supposes > 0 is the first value such that (3.11) holds for all nonnegative t ≤ s while Us isnot transversal to Ps at a point p ∈ ∂Us. By the local convexity of M , Ps is a localsupporting hyperplane to M at p where M locally lies in the half space (z−q)·νP ≤ s.For ε > 0 small enough,

Vε := M ∩p (z − p) · (−νP ) ≤ ε

is transversal to Ps−ε and is a convex disk. Moreover, ∂Vε = ∂Us−ε, for ∂Us−ε is aglobally convex disk as its boundary is contained in a hyperplane (see [17]). Thisimplies M = Vε ∪ Us−ε and therefore is a closed convex sphere without boundary,which is a contradiction.

We now return to the proof of Theorem 3.1. Let t0 > 0 be the smallest value suchthat Ut0 ∩ ∂M 6= ∅ and choose a point p ∈ Ut0 ∩ ∂M . Note that U := Ut0 is globallyconvex. We consider the special coordinate system at p which satisfies (3.4). Underthis coordinate system, q lies in the region |xn+1| ≤ xn cotβ. In particular, xn(q) > 0.We also note that

νP = en cos θ + en+1 sin θ

for some θ ∈ [β, π − β]. Moreover, M locally (near p) is given as the graph of afunction u on a domain Ωp as in (3.9) satisfying (3.1).

Let r > 0 be as in (3.9). We see from above that if xn(q) < r/2 then q is on thegraph of u over Ωp and we are done. So we next consider the case that xn(q) ≥ r/2.Let Cq = Cq(∂U) be the convex cone generated by ∂U with vertex q. We will showthat Cq contains a non-degenerate cone of fixed size that contains p. This meansthere exists a point q0 ∈ R

n+1, |q0 − q| = 1, and a uniform constant δ0 > 0 such thatp ∈ Cq(Bδ0(q0)) ⊂ Cq where Cq(Bδ0(q0)) is the cone generated by Bδ0(q0) with vertexq. Since |q − p| ≥ r/2, this will complete our proof under assumption (3.11).

Choose new coordinates (y1, . . . , yn+1) in Rn+1 with origin at p such that yi = xi

(1 ≤ i ≤ n − 1), yn(q) > 0, yn+1(q) = 0 and let τi denote the unit vector in thepositive yi direction (0 ≤ i ≤ n+ 1). We have yn(q) ≥ xn(q) ≥ r

2 and hence

(τk · (q − p))2

|q − p|2=

(ek · (q − p))2

|q − p|2≤ 1−

( r

2dM

)2∀ 1 ≤ k ≤ n− 1.(3.12)

From the convexity of U we see that Cq contains the cone generated by Γ(p) withvertex q since Γ(p) and q are separated by the hyperplane containing ∂U . By (3.6),(3.7) and (3.12) the projection of Cq to the hyperplane R

n ≡ yn+1 = 0 contains


an n-ball Bρ(0) in Rn where ρ ≥ c0 for a uniform constant c0 > 0. To complete the

proof, therefore, we only have to find a point p0 with

yn+1(p0) > 0 andyn(q)− yn(p0)yn+1(p0)

≤ C0(3.13)

for some uniform constant C0 > 0 such that the cone generated by the convex hull ofΓ(p) ∪ p0 with vertex q is contained in Cq. (We note that it is always possible tofind such p0 on Γ(p) with C0 = C0(t0) depending on t0; C0(t0) may, however, tend toinfinity as t0 → 0.)

For 0 ≤ t ≤ yn(q), letWt = yn ≥ t∩qM . IfWyn(q)∩∂U 6= ∅ then we are done since∂U lies in the upper half space yn+1 ≥ 0. We thus may assume Wyn(q) ∩ ∂U = ∅.Note that then Wyn(q) ⊂ U and is therefore a convex cap. We may find t1 ∈ [0, yn(q))such that Wt ∩ ∂M = ∅ for all t1 < t ≤ yn(q) and Wt1 ∩ ∂M 6= ∅. Note that Wt1 isalso a convex cap and Wt1 \ U ⊂ Cp by convexity.

If t1 = 0 then

τn ·Π(X,X) ≤ 0, ∀ X ∈ Tp∂M(3.14)

since Γ(p) lies in the half space yn ≤ 0. (This implies that Γ(p) is contained in thehalf space xn+1 ≥ 0.) It follows from (3.5) that τn+1 · en ≥ sinβ, that is, the anglebetween τn+1 and en does not exceed π

2 − β. Consequently,

yn+1(z) = z · τn+1 ≥ xn(z)en · τn+1 ≥ xn(z) sinβ ∀ z ∈ Γ(p).

since xn+1(z) ≥ 0. We see any point p0 on Γ(p) with xn(p0) ≥ r2 must satisfy (3.13).

We now assume t1 > 0 and take an arbitrary point p1 ∈ Wt1 ∩ ∂M . We havep1 ∈ (Wt1 \ U) ∪ ∂U ⊂ Cq and, similarly to (3.14),

τn ·X = 0 and τn ·Π(X,X) ≤ 0, ∀ X ∈ Tp∂M.(3.15)

Moreover, since Wt1 ∩ yn+1 ≤ 0 ⊂ U ,

yn+1(p1) > 0 and 0 < yn(p1) = t1 < yn(q).(3.16)

We may further assume that there exists a uniform constant ε0 > 0 such that

|X · τn+1| ≤ ε0(3.17)

and

|Π(X,X) · τn+1| ≤ ε0(3.18)

for all unit tangent vector X to ∂M at p1. This can be seen as follows. Suppose thereis a unit vector X ∈ Tp1∂M which does not satisfy (3.17) or (3.18) and let γX be the


geodesic on Γ(p1) tangential to X at p0. We can then find a point p0 ∈ γX near p1

such that, if (3.17) is violated then (3.13) holds for C0 = C0(ε0), while if (3.18) fails,

|Y · τn+1| ≥ ε1

for some unit tangent vector Y to ∂M at p0 and some uniform constant ε1 > 0.Note that (3.17) and (3.18) imply

νΣ(p1) · τn ≤ 0(3.19)

when ε0 is sufficiently small, since the angle between Π(X,X) and −τn is sufficientlysmall while that between Π(X,X) and νΣ(p1) does not exceed π

2 −2β. By (3.16) and(3.19) we obtain

νΣ(p1) · τn+1 ≤ 0,(3.20)

since the segment joining p1 and q locally lies on the inner side of Σ near p1. Finally,by (3.19), (3.20) and the local strict convexity of Σ near boundary there exists a pointz ∈ Σδ ∩ V with

yn+1(p1)− yn+1(z) ≥ c0

for some uniform constant c0 > 0 depending on δ and κmin[Σδ], where V is the vertical2-plane (in y-coordinates) through p1 and q. Since z must lie above the line throughq and p1 by the convexity of M and the assumption that Σδ does not intersect M ininterior, we have yn+1(z) ≥ yn+1(q) = 0. Thus yn+1(p1) ≥ c0 and p0 := p1 satisfies(3.13) where C0 > 0 depends on δ and κmin[Σδ].

Step 3. We now assume there is no hyperplane through q satisfying assumption(3.11). We will first prove that M has a unique local supporting hyperplane (thus atangent hyperplane) at q.

Let P be a local supporting hyperplane at q to M and let E denote the set ofpoints on ∂M that (intrinsically) belong to P ∩q M . Clearly E 6= ∅. We claim thatq is contained in the convex hull of E. Indeed, if this is not the case, that is, q andE are separated by a hyperplane, we may assume P = xn+1 = 0 and q lies in theregion xn > ε while E in xn < −ε for some ε > 0. Then M ∩q xn+1 ≤ axn doesnot intersect ∂M where a > 0 is sufficiently small, which is a contradiction.

By Caratheodory’s theorem (cf. [22]) q is contained in an l-dimensional simplex Swith vertices in E for some 1 ≤ l ≤ n. We have S ⊆ P ∩q M by the local convexityof M .


Let p be a vertex of S and consider the special coordinate system at p. We notethat, by the local convexity of M , P is a local supporting hyperplane to M at everypoint on the segment pq joining p and q. It follows that

νP = en cos θ + en+1 sin θ(3.21)

for some θ ∈ [β, π−β]. Recall M locally near p is the graph of a convex function overa domain Ωp in xn+1 = 0. Since Ωp is strictly convex the segment pq is transversalto ∂M at p. For otherwise pq would be contained in xn+1 = 0 and tangential to∂Ωp at p, resulting in a contradiction as Ωp would contain points on pq other thanp. Consequently, P is the tangent hyperplane to M at p as P contains pq and istangential to ∂M at p.

Next, assume furthermore that pq ⊂ pp1 ⊆ S for some p1 6= q. Let Q be a localsupporting hyperplane to M at q. Then pp1 ⊂ Q and therefore Q is a local supportinghyperplane to M at every point on pp1. We have Q = P since both are the tangenthyperplane to M at p. This also shows that P is the tangent hyperplane of M atevery point on pp1 (except possibly p1). Consequently, u extends along pp1.

As we can always find a point p ∈ E such that the segment pq extends in S, wehave proved the uniqueness of the local supporting plane to M at q. Using inductionon l we will next prove the assertion in the Theorem at point q.

Let us first consider the case l = 1, that is, S = pp1 where p, p1 ∈ ∂M . Suppose|p − q| ≤ |p1 − q| and let Ω be the convex hull (in R

n = xn+1 = 0) of p′1 ∪ Ωp

where p′1 ∈ Rn with p1 = (p′1, xn+1(p1)). (Similar meaning for q′ below.) As in Step 2

we may assume xn(q) ≥ r2 where r as in (3.9). This implies (3.12), that is the angle

between pq and ek has a uniform positive lower bound for all 1 ≤ k ≤ n− 1.Let

v(x) = supL(x) : L is an affine function, L ≤ u at p′1 and on Ωp, x ∈ Ω.(3.22)

Then v is a convex function and detD2v = 0 in Ω. We have u ≤ v where u is definedin Ω. Since |p1 − q| ≥ |p − q| ≥ r

2 , by (3.12) there exists a uniform constant λ > 0depending on r and max∂M |Π|, such that the n-ball Bλ(q′) is contained in Ω. Bythe local convexity of M we see u is defined on Bλ/2(q′) ⊂ Ω with a uniform boundon ‖u‖C0,1(Bλ/2(q′)). This completes the proof for l = 1.

Assume now l > 1 and suppose we have proved the assertion for any point in asimplex of dimension less than l with vertices in E. Choose p ∈ E and p1 on an (l−1)dimensional face of S such that q ∈ pp1. If |p − q| ≤ |p1 − q| then the proof followsas exactly in case l = 1. Let us therefore assume |p − q| ≥ |p1 − q|. By induction,


in a suitable coordinate system (y, yn+1), y ∈ Rn with origin at p1, M locally near

p1 is the graph of a convex function yn+1 = u(y) with a uniform C0,1 bound in ann-ball BR(0) where R is a uniform constant. Since P is the tangent hyperplane toM at any point on pp1 (except possibly p1), we have νP · (0, . . . , 0, 1) ≥ c0 for someuniform constant c0 > 0. Thus u extends along pp1. Replacing the convex function vin (3.22) by

v(y) = supL(y) : L is an affine function, L ≤ u at p and on BR(0),(3.23)

defined in the convex hull of p ∪BR(0), the rest of proof follows that of case l = 1.This, finally, completes our proof.

An important consequence of Theorem 3.1 is a compactness result (Theorem 3.4)which we will need in the next section. First, it follows immediately from Theorem 3.1that

Corollary 3.3. There exist uniform constants R, r > 0 depending on δ, κmin[Σδ],κmax[Σδ], max∂Σ |Π| and dM such that for any p ∈M , M ∩p BR(p) is embedded andthe convex body in BR(p) bounded by M ∩p BR(p) contains a ball of radius r.

According to a convergence theorem of Alexander-Ghomi [1] we thus have

Theorem 3.4. Let Mk be a sequence of locally convex hypersurfaces contained ina bounded region in R

n+1 with ∂Mk = ∂Σ for all k. Suppose each Mk lies on theinner side of Σ and does not intersect Σδ. Then there exists a subsequence Mkiconverging in Hausdorff metric to a locally convex hypersurface M with ∂M = ∂Σ.Moreover, for each i there exists a homeomorphism form Mki on to M with boundaryfixed.

Proof. We refer to [1] (Theorem 7.1) for the major part of the proof. Here we onlypoint out that by Corollary 3.3 the conditions of Theorem 7.1 in [1] are satisfied, andgive a brief proof of the fact ∂M = ∂Σ and that M is C0,1 up to the boundary. Givena point p ∈ ∂Σ, we consider the special coordinates at p satisfying (3.4). Then eachΣk locally near p can be represented as a convex graph xn+1 = uk(x) over a domainΩp of form (3.9) with a uniform C0,1 norm bound. By compactness there exists asubsequence of uk converging to a convex function u ∈ C0,1(Ωp). Moreover, wehave u = u on x ∈ ∂Ωp : xn = ϕ since u ≤ uk ≤ v in Ωp where v is as in (3.8), ϕ asin (3.6) and the graph of u represents Σ. Note that M must coincide with the graphof u near p. Consequently, M is a locally convex hypersurface of class C0,1 up to theboundary and ∂M = ∂Σ.


We next derive a priori bounds for all principal curvatures for smooth locallystrictly convex K-hypersurfaces.

Theorem 3.5. Assume in addition that M is a smooth locally strictly convex hyper-surface of constant Gauss curvature K > 0. Then

1C2≤ κmin[M ] ≤ κmax[M ] ≤ C2

where C2 > 0 depends on K, K−1, δ, κmin[Σδ], κmax[Σδ], max∂Σ |Π|, and dM .

Proof. We first establish the estimate on the boundary. Given any point p ∈ ∂M ,by Theorem 3.1 we may write M locally (near p) as a graph xn+1 = u(x) with ana priori gradient bound over a smooth strictly convex domain Ωp where u satisfiesthe Gauss curvature equation (2.1). As ∂Ωp is strictly convex we may appeal to theboundary estimates for |∇2u| due to Caffarelli-Nirenberg-Spruck [6] (which is local innature) to obtain

|uij(0)| ≤ C(3.24)

where C depends on ‖u‖C1(Ωp) and geometric quantities of Σδ and ∂Σ. Since theprincipal curvatures of M at p are the eigenvalues of the matrix

(1 + |∇u|2)−12uij

(with respect to δij+uiuj, the metric of M), the desired estimates follow from (3.24)and the fact that the Gauss curvature is the product of all principal curvatures.

Turning to the global estimates, consider Λ := maxκeρ where

ρ(x) = |x− x0|2, x ∈ Rn+1

(x0 is a fixed point in Rn+1), and the maximum is taken for all normal curvatures κ

over M . As we already have estimates for principal curvatures on ∂M , we may assumeΛ is attained at an interior point p ∈ M . Choose coordinates in R

n+1 with origin atp such that the tangent hyperplane of M at p is given by xn+1 = 0 and M locally iswritten as a strictly convex graph xn+1 = u(x) where x = (x1, . . . xn) ∈ R

n. We mayalso assume the Hessian matrix uij to be diagonal at 0 with u11(0) ≥ uii(0) > 0for all 1 < i ≤ n. Note that, since Du(0) = 0, uii(0) (1 ≤ i ≤ n) are the principalcurvatures of M at p. Thus Λ is achieved at p with respect to the normal curvaturein x1 direction which is locally given by

κ =u11

(1 + u21)w

, w = (1 + |∇u|2)12 .


Since the function log u11 − log(1 + u21) − logw + ρ then has a maximum at the

origin where Du = 0, w = 1, Dw = 0 and wii = u2ii for all 1 ≤ i ≤ n, we have at 0,

u11i

u11− wiw− 2u1u1i

1 + u21

+ ρi = 0, 1 ≤ i ≤ n(3.25)

andu11ii

u11− (

u11i

u11)2 − u2

ii − 2u21i + ρii ≤ 0, 1 ≤ i ≤ n.(3.26)

Multiplying (3.26) by u11/uii and taking sum over i from 1 to n, one obtains∑ u11ii

uii−∑ (u11i)2

u11uii− u11∆u− 2u2

11 +∑ ρii

uii≤ 0.(3.27)

Differentiating equation (2.1) we have for 1 ≤ k ≤ n,

uijuijk = (n+ 2)wkw

and

uijuijkk − uilujmuijkulmk = (n+ 2)wkkw− (n+ 2)

w2k

w2,

where uij is the inverse matrix of uij Combining these and (3.27) we obtain

nu211 − u11∆u+

∑ ρiiuii≤ 0.(3.28)

Next,

ρ(x) = |x− x0|2 + (u(x)− x0n+1)2, x ∈ R

n

where x0 = (x0, x0n+1), and therefore,

ρii = 2 + 2(u(x)− x0n+1)uii + u2

i .

Since ∆u ≤ nu11, by (3.28) one sees that at 0,

0 ≥n∑i=1

ρiiuii≥

n∑i=2

2uii− 2nx0

n+1

≥ 2

(u22 · · ·unn)1

n−1

− 2nx0n+1

≥ 2( u11

detuij

) 1n−1 − C

= 2(u11

K

) 1n−1 − C.

It follows that

u11(0) ≤ CK.


This proves an upper bound for κmax[M ], from which a lower bound for κmin[M ] canbe derived in terms K−1. The proof is complete.

Remark 3.6. Using an estimate of Guan-Trudinger-Wang [14] in place of that of [6],it is possible to obtain an upper bound for the principal curvatures which does notdepend on the lower bound of Gauss curvature.

4. Deformation to K-hypersurfaces

The primary purpose of this section is to prove the existence part in Theorems 1.1and 1.2. Throughout the section, let Σ be a locally convex immersed hypersurface inRn+1 with embedded boundary ∂Σ and Gauss curvature KΣ ≥ K everywhere on Σ,

where K is a fixed non-negative constant. Our idea is to deform Σ to a locally conveximmersed hypersurface M with KM ≡ K and ∂M = ∂Σ.

Let D ⊆ Σ be a disk on Σ which, as a hypersurface in Rn+1, may be represented

as the graph of a convex function u defined in a domain Ω (in some hyperplane) withLipschitz boundary. By Theorem 2.1, there is a unique function u ∈ C0,1(Ω) whosegraph is a convex hypersurface D of constant Gauss curvature K with ∂D = ∂D. Bythe maximum principle, we have u ≥ u in Ω. Thus D lies on the inner side of D.

This naturally induces a C0,1-diffeomorphism ΨD : Σ→ Σ := D ∪ (Σ \D) which isfixed on Σ \ D. The hypersurface Σ is locally convex with KΣ ≥ K and ∂Σ = ∂Σ.We call Σ a basic lifting of Σ (by D over D). A lifting of Σ is a hypersurface which isobtained by a finite number of basic liftings starting from Σ. We introduce a partialorder between liftings of Σ: Σ1 Σ2 if and only if Σ2 is a lifting of Σ1 or Σ2 = Σ1.

Lemma 4.1. Let Σ1 and Σ2 be any two liftings of Σ. Then there exists a uniquelifting, which we denote as Σ1 ∨Σ2, of Σ such that Σ1 Σ1 ∨Σ2, Σ2 Σ1 ∨Σ2, andΣ1 ∨ Σ2 N for any lifting N with Σ1 N and Σ2 N .

Proof. We first assume Σ1 is a basic lifting of Σ by D1 over a disk D1 ⊆ Σ and letA be the open region in R

n+1 bounded by D1 ∪ D1. Assume Σ2 to be a lifting of Σover a region D2. (D2 is not necessarily a disk.) Intuitively, if Σ, Σ1 and Σ2 are allembedded, then it is obvious that the hypersurface

Σ1 ∨ Σ2 := (Σ2 \ (Σ2 ∩A)) ∪ (D \ (D ∩B))

where B is the open regions in Rn+1 bounded by Σ2 ∪ Σ, is a lifting of Σ with the

desired properties. In the general case when some of these hypersurfaces may be


immersed, we view Σ as an immersion

Φ0 : Σ0 → Σ ⊂ Rn+1(4.1)

of a differentiable manifold Σ0 and let

Φi : Σ0 → Σi ⊂ Rn+1, i = 1, 2(4.2)

be the immersions induced from the liftings. (Note that Φi = Φ0 on Σ0 \ Φ−10 (Di).)

The lifting Σ1 ∨ Σ2 is then given by the immersion

Φ : Σ0 → Σ1 ∨ Σ2 := Φ(Σ0) ⊂ Rn+1

defined as

Φ(p) :=

Φ1(p), if p ∈ Φ−1

0 (D1) \ Φ−10 (D2),

Φ1(p), if p ∈ Φ−10 (D1) ∩ Φ−1

0 (D2) and Φ2(p) ∈ A,Φ2(p), otherwise,

(4.3)

for p ∈ Σ0. The general case now can be proved by induction.

The next lemma, which states that volume decreases under lifting, is well known;for completeness we include a proof.

Lemma 4.2. Let Σ1 and Σ2 be liftings of Σ. If Σ1 Σ2 then Vol(Σ1) ≥ Vol(Σ2).Moreover, the equality holds if and only if Σ1 = Σ2.

Proof. Obviously we may assume Σ2 is a basic lifting of Σ1 over a disk D1 ⊂ Σ1.Suppose D1 and its lifting D2 ⊂ Σ2 are the graphs of convex functions u1 and u2 overa domain Ω ⊂ R

n, respectively. We have u1 ≤ u2 on Ω and u1 = u2 ∂Ω.Let

N(x, z) =(∇u2,−1)√1 + |∇u2|2

, (x, z) ∈ Ω× R.

denote the downward unit normal vector to D2 at (x, u2(x)). Thus divN(x, z), thedistributional mean curvature of D2 at the point (x, u2(x)) with respect to the upwardnormal vector, is nonnegative almost everywhere since u2 is a convex function. Let

ω = (x, z) ∈ Rn+1 : u1(x) < z < u2(x), x ∈ Ω.

By the divergence theorem we have

0 ≤∫ω

divNdv =∫D1

N · ν1dσ −∫D2

dσ

= Vol(D1)−Vol(D2) +∫D1

(N · ν1 − 1)dσ


where ν1 is the downward unit normal vector to D1. Since 0 ≤ N · ν1 ≤ 1 on D1 wehave Vol(D1)−Vol(D2) ≥ 0; obviously, the equality holds only when D1 = D2.

We need one more lemma which states that volume is continuous under uniformconvergence of uniformly Lipschitz convex functions.

Lemma 4.3. Let wk be a sequence of uniformly Lipschitz convex functions on Ωconverging uniformly to w. Then∫

Ω

√1 + |∇w|2 dx = lim

k→∞

∫Ω

√1 + |∇wk|2 dx(4.4)

Proof. Let Wk =√

1 + |∇wk|2 ; then |∇Wk| ≤ |∇2wk| ≤ ∆wk a.e. in Ω since wk isconvex. Therefore, ∫

Ω|∇Wk| dx ≤ sup

Ω|∇wk||∂Ω| .

Hence Wk are uniformly bounded in W 1,1 and so converge in L1 to√

1 + |∇w|2.

We are now ready to prove the main result of this section. Let L be the collectionof liftings of Σ and set

µ := infL∈L

Vol(L).

Theorem 4.4. Suppose Σδ is C2 and locally strictly convex up to the boundary forsome fixed δ > 0. There exists a locally convex hypersurface M in R

n+1 of class C0,1

up to the boundary with ∂M = ∂Σ and KM ≡ K. Moreover, M is homeomorphic toΣ and Vol(M) = µ.

Proof. For each k ≥ 1 choose Σk ∈ L such that

Vol(Σk) ≤ µ−1k.

By Lemmas 4.1 and 4.2 we may assume Σk Σk+1 for all k ≥ 1. According to The-orem 3.4 after passing to a subsequence we may assume Σk converges in Hausdorffmetric to a locally convex hypersurface M which, in addition, is homeomorphic toeach Σk. Clearly ∂M = ∂Σ. It remains to show Vol(M) = µ and KM ≡ K.

Consider a point p ∈M . There exists a sequence pk ∈ Σk, k = 1, 2, . . . , convergingto p (in R

n+1) such that Σk ∩pk BR(pk) converges to M ∩pBR(p) in Hausdorff metricwhere R > 0. According to Theorem 3.1, when R is chosen sufficiently small eachΣk∩pkBR(pk) can be represented as the graph of a convex function wk with a uniformC0,1 norm bound (independent of k). By compactness we may choose a coordinate


system in Rn+1 such that, after possibly passing to subsequences, all the functions

wk are defined in a fixed domain Ω ∈ Rn satisfying

‖wk‖C0,1(Ω) ≤ C0 independent of k(4.5)

and wk converges uniformly to a function w ∈ C0,1(Ω) whose graph obviously locallyrepresents M . Hence by Lemma 4.3 and a covering argument, Vol(M) = µ.

Consider now the Dirichlet problem for the Gauss curvature equation (2.1) in Ω.Using wk as a subsolution for each k ≥ 1, by Theorem 2.1 we obtain a unique convexsolution uk ∈ C0,1(Ω) of (2.1) satisfying uk = wk on ∂Ω. We have uk ≥ wk on Ω andby (4.5)

‖uk‖C0,1(Ω) ≤ C0 independent of k.

Thus there exists a subsequence, which we still denote by uk, converging to a convexfunction u in C0,1(Ω). We see u satisfies (2.1) and u ≥ w on Ω with u = w on ∂Ω.

On the other hand, for each k ≥ 1 let Σk be the lifting of Σk obtained by replacingDk with Dk, where Dk and Dk are the graphs of wk and uk over Ω, respectively.Similarly, let M be the locally convex hypersurface obtained from M by replacing thegraph of w over Ω by that of u. Clearly Σk converges to M as uk converges uniformlyto u on Ω. Since by Lemma 4.2 µ ≤ Vol(Σk) ≤ Vol(Σk) for each k it follows thatVol(M) = µ and therefore Vol(D) = Vol(D). As both u and w are convex functions,this implies u ≡ w on Ω by the proof of Lemma 4.2. Since u satisfies (2.1), M hasconstant Gauss curvature K in a neighborhood of p.

5. Regularity

In this section we study the regularity of the hypersurface M constructed in theprevious section to complete our proof of Theorems 1.1 and 1.2. Throughout thissection, we assume, as in section 4, that Σ is a locally convex immersed hypersurfacewhich is C2 and locally strictly convex along its boundary ∂Σ. Thus Σ is C2 andlocally strictly convex in a neighborhood of, and up to, ∂Σ. In addition, we assume∂Σ to be embedded and smooth. Let K ≤ minKΣ be a non-negative constant andlet M the locally convex hypersurface with KM ≡ K and ∂M = ∂Σ constructed insection 4. By Theorem 4.4, M is C0,1 up to the boundary.

Theorem 5.1. If K > 0 then M is smooth up to the boundary and locally strictlyconvex.


Proof. Consider an interior point p ∈ M which we assume to be the origin of Rn+1.

Since M is of class C0,1, M locally near p can be represented as a convex graphxn+1 = u(x) ≥ 0 over a domain Ω1 ⊂ R

n ≡ xn+1 = 0 with a C0,1 norm bound

‖u‖C0,1(Ω1) ≤ C1.

It follows that u satisfies the inequalities in the viscosity sense

K ≤ det(uij) ≤ K(1 + C21 )

n+22 in Ω1.

We may assume M ∩p xn+1 = 0 ⊂ Ω1. By a theorem of Caffarelli [2], the nodal setu = 0 either is a single point, in which case M is smooth and strictly convex at p(see [4]), or does not contain any interior extreme points. So we will be done if we canshow that u = 0 = 0. Suppose this is not the case. Then we can find two pointsq1, q2 ∈ ∂M such that q1q2 ⊆ M ∩ xn+1 = 0 and xn+1 = 0 is a local supportingplane of M at every point on q1q2. By the proof (Step 3) of Theorem 3.1, q1q2 istransversal to ∂M at the endpoints. Without loss of generality we may assume

qi = (0, . . . , 0, (−1)ia, 0), i = 1, 2,

where a > 0. Consequently, there exists a constant δ > 0 such that, in a neighborhoodof q1q2, M is given as a convex graph xn+1 = u(x) ≥ 0 over a domain

Ω0 := x := (x′, xn) ∈ Rn|ϕ1(x′) < xn < ϕ2(x′) for |x′| < δ

where ϕ1, ϕ2 are smooth functions since ∂M is smooth and transversal to q1q2. Letψ be a smooth function defined on ∂Br, where Br ⊂ Ω0 is the n-ball of radius r ≤ δ

centered at the origin, satisfying ψ(0,±r) = 0 and

ψ(x′, xn) ≥ maxu(x′, ϕ1(x′)), u(x′, ϕ2(x′)), ∀ (x′, xn) ∈ ∂Br.

This is possible since both u(x′, ϕ1(x′)) and u(x′, ϕ2(x′)) are smooth in x′ as ∂Mis smooth and tangential to xn+1 = 0. By [6] there exists a unique strictly convexsolution v ∈ C∞(Br) to the Dirichlet problem of the Monge-Ampere equation

det(vij) = K in Br, v = ψ on ∂Br.

Since det(vij) = K ≤ det(uij) in Br and, by the convexity of u,

u(x′, xn) ≤ maxu(x′, ϕ1(x′)), u(x′, ϕ2(x′)) ∀ (x′, xn) ∈ Ω0.

which implies v ≥ u on ∂Br, by the comparison principle we have v ≥ u ≥ 0 on Br. Bythe strict convexity of v, however, we have v(0) < 0 since v(0, a) = v(0,−a) = 0, whichis a contradiction. This proves that M is strictly convex and smooth in any interiorpoint, while the boundary regularity follows from [6]. The proof is complete.


This completes the proof of Theorem 1.1. Turning to the case K = 0 we first provethe following lemma.

Lemma 5.2. Let N be a locally convex hypersurface with KN ≡ 0. Let p be aninterior point of N and P a local supporting hyperplane to N at p. Then p is containedin a k-dimensional sub simplex of N ∩p P with vertices on ∂N for some 1 ≤ k ≤ n.

Proof. This follows form the argument in Step 3 of the proof of Theorem 3.1 as thereis no hyperplane through p satisfying assumption (3.11). We redo the proof here forthe reader’s convenience. Since N is locally convex, P is a local supporting hyperplaneto N at every point on N ∩p P . Let D be the set of points on ∂N that (intrinsically)belong to N ∩p P . It suffices to show that any point in N ∩p P is contained in theconvex hull of D. If this is not the case, there is a point q ∈ N ∩pP which is separatedby a hyperplane from D. We may assume P = xn+1 = 0 and q lies in xn > ε whileD lies in xn < −ε for some ε > 0. It then follows that N ∩q xn+1 < δxn is containedin the interior of N when δ is sufficiently small. This is a contradiction as the Gausscurvature of N ∩q xn+1 < δxn is zero everywhere while its boundary is containedin the hyperplane xn+1 = δxn.

Theorem 5.3. If K = 0, then M is C1,1 up to the boundary.

Proof. Let p be an interior point of M . From Step 3 of the proof of Theorem 3.1 wesee that M has a tangent hyperplane at p. Suppose M locally (near p) is writtenas a convex graph xn+1 = u(x) with u ≥ 0 over TpM := xn+1 = 0. Since TpM isthe tangent hyperplane to M at every point on M ∩p TpM , u is defined in a domainΩ ⊂ R

n such that M ∩p TpM ⊂ (x, 0) : x ∈ Ω. By [7], in order to prove that M isC1,1 it suffices to show that there exists a constant C, depending only on ∂M , andε = ε(p) > 0 such that

u(x) ≤ C|x|2(5.1)

for all x ∈ Bε(x0) ⊂ Rn where p = (x0, 0).

By Lemma 5.2, p is contained in a k-dimensional subsimplex, which we denote asS, of M ∩p TpM with vertices on ∂M for some 1 ≤ k ≤ n. According to [7], in orderto prove (5.1) it suffices to consider the case k = 1. Suppose now that S is a segmentwith end points q1 := (x1, 0), q2 := (x2, 0) on ∂M . By the proof of Theorem 3.1,S is transversal to ∂Ω at the end points and both ∂Ω and u|∂Ω are smooth in aneighborhood of xi (i = 1, 2). Of the two end points, suppose that q2 is the closer top. We may assume x2 = 0 and en to be the interior unit normal to ∂Ω at 0 where ek


(1 ≤ k ≤ n+ 1) is the unit vector in the positive xk-axis direction. Since xn+1 = 0 isa local supporting hyperplane to M at q2, en+1 ·Π(X,X) ≥ 0 for any X ∈ Tq2∂M .On the other hand, from the proof of Theorem 3.1 we see that the angle betweenΠ(X,X) and νΣ at q2 does not exceed π

2 − 2β for some uniform constant β > 0. Itfollows that

νΣ(q2) = en cosα+ en+1 sinα(5.2)

where 2β − π2 ≤ α ≤

π2 . We distinguish two cases: (i) α ≤ β and (ii) α > β.

If α ≤ β, then for any X ∈ Tp∂M , the angle between Π(X,X) and en is less than orequal to π

2 −β and, therefore, en ·Π(X,X) ≥ sinβ > 0. This implies that ∂Ω∩Bδ(0)is uniformly strictly convex where δ > 0 is a uniform constant. We therefore mayfollow the proof of [7] to derive (5.1).

We now suppose α > β. Then locally Σ is a strictly convex graph xn+1 = u(x)over Ω ∩ Bδ(0) for some uniform constant δ > 0. To prove (5.1) we then can followthe proof of Theorem 3.2 in [11]. This proves Theorem 5.3.

Remark 5.4. Theorem 1.2 follows from Lemma 5.2 and Theorem 5.3. If minKΣ > 0,Theorem 5.3 may be proven by approximation as follows. For any positive constantε ≤ minKΣ, by Theorem 5.1 there exists a smooth locally strictly convex hyper-surface M ε with constant Gauss curvature ε and ∂M ε = ∂Σ. By Theorem 3.5 (seeRemark 3.6) we obtain a subsequence εk → 0 such that M εk is convergent in localC1,1 norms. Clearly, the limiting hypersurface must be M . Consequently, M is C1,1

up to the boundary.

Remark 5.5. We have the following characterization of M at boundary (for K = 0).Let p ∈ ∂M and choose coordinates of R

n+1 with origin at p such that en+1 and en

are the unit normal and interior conormal to TpM , respectively, where ek as before isthe unit vector in the positive direction of xk axis.

Proposition 5.6. Let K = 0 and p ∈M . Suppose M ∩p TpM does not contain anypoint in ∂M ∩ xn > 0. Then there exists some unit vector X ∈ Tp∂M such thatθ(X) = 0 where θ(X) is defined by

Π(X,X) = |Π(X,X)|(en cos θ(X) + en+1 sin θ(X)), X ∈ Tp∂Σ, X 6= 0.(5.3)

Proof. Suppose

minθ(X) : X ∈ Tp∂Σ, |X| = 1 > 0.


Then M ∩p 0 ≤ xn+1 ≤ λxn does not contain any point on ∂M ∩ xn > 0 whenλ > 0 is sufficiently small. By Lemma 5.2 this implies

(M ∩p 0 ≤ xn+1 ≤ λxn) ∩ xn > 0 = ∅,

contradicting the fact that TpM = xn+1 = 0.

6. Locally convex hypersurfaces with extreme boundary

We first give a brief proof of Corollary 1.3.

Proof of Corollary 1.3. By Theorem 1.2 we obtain a globally convex hypersurfaceM with KM ≡ 0 and ∂M = Γ. Moreover, M is on the inner side of Σ along theboundary. Consider now an arbitrary interior point q ∈ Σ and let P be a localsupporting hyperplane to Σ at q. Since Σ is C2 and locally strictly convex near theboundary, Σ ∩q P does not (intrinsically) contain points on ∂Σ. Let t0 > 0 be thesmallest value such that Σt0 contains a point p on ∂Σ, where

Σt := Σ ∩q z ∈ Rn+1 : (z − q) · νP ≤ t, t ≥ 0.

We see that ∂Σ locally near p lies in the half space z ∈ Rn+1 : (z−p) ·νP ≥ 0. Since

M is globally convex, ∂M lies on one side of TpM . Let us assume TpM = xn+1 = 0and that ∂M lies in xn+1 ≥ 0.

Choose coordinates in Rn+1 such that en+1 and en being the unit normal and

interior conormal to ∂M at p, respectively. We claim that

νP = en cosα+ en+1 sinα, for some 0 ≤ α ≤ π2 .(6.1)

Note that this implies Σt0 ⊂ xn+1 ≤ 0 and thus completes the proof.For any 0 < t < t0, since Σt does not contain points on ∂M , it is easy to see

that (Σ \ Σt) ∪ Dt is a lifting of Σ with respect to K = 0, where Dt is the regionon the hyperplane Pt := z ∈ R

n+1 : (z − q) · νP = t bounded by Σt ∩ Pt. ThusM ∩p z ∈ R

n+1 : (z − q) · νP ≤ t0 does not intersect the region bounded by Σt ∪Dt

for any 0 < t < t0. Consequently, TpM does not intersect the interior of Σt0 . Thisproves (6.1).

We next construct a smooth locally strictly convex, non-embedded, surface M ofGauss curvature one in R

3 such that ∂M is strictly extreme. Let S1 be the unit spherecentered at (1

2 , 0, 0). Cut off a small cap from the top of S1 using a plane perpendicularto the line through (0, 0, 2) and the center of S1. Let Σ1 be the resulting spherical capand Σ2 the reflection of Σ1 with respect to x1 = 0. Now, connecting the boundary


circles of Σ1 and Σ2 by a thin convex bridge, we obtain a locally strictly convexsurface Σ with self-intersection. Moreover, ∂Σ is strictly extreme. According to thebridge principle of Hauswirth [16] there exists a locally strictly convex surface M ofconstant Gauss curvature one with the same boundary. It follows from [16] that Mis a small perturbation of Σ and therefore has self-intersection.

If we start with cutting a small cap from the top of S1 using a horizontal planeand repeat the rest of the above procedure, we get a non-embedded locally strictlyconvex K-surface M such that ∂M is extreme while the boundary of the convex hullof ∂M has interior singularities along the bottom edges.

References

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Geom., 7 (1999), 497–550.[17] J. van Heijenoort, On locally convex manifolds, Comm. Pure. Appl. Math., 5 (1952), 223-242.


[18] D. Hoffman, H. Rosenberg and J. Spruck, Boundary value problems for surfaces of constantGauss curvature, Comm. Pure Applied Math., 45 (1992), 1051-1062.

[19] J.-X. Hong, The estimates near the boundary for solutions of Monge-Ampere equations, J. PartialDiff. Equations, 7 (1994), 97-110.

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[22] P. J. Kelly and M. L. Weiss, Geometry and Convexity, John Wiley & Sons, New York-Chichester-Brisbane, 1979.

[23] N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations in a domain, Izv.Akad. Nauk. SSSR Ser. Mat., 47 (1983), 75–108 (Russian); English translation in Math. USSRIzv., 22 (1984), 67-97.

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Ann. Inst. Henri Poincare -Anylyse Non Linearire, 8 (1991), 499-522.

Department of Mathematics, University of Tennessee, Knoxville, TN 37996, Fax:

(865) 974-6576.

E-mail address: [email protected]

Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, Fax:

(410) 516-5549.

E-mail address: [email protected]

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